Iterated edge drawer animation nodes
A spiral or a helix is described using the set of parametric equations as follows:

$$
\begin{aligned}
x &amp;= R\cos{t}\\
y &amp;= R\sin{t}\\
z &amp;= \delta t
\end{aligned}
$$

Where $R$ is the radius, and $\delta$ is the vertical frequency of the curve. By implementing this, we get:



Now, we are going to move the points of the helix with a random amount in the direction of your choice, I can see you are offsetting them in the direction of the $x$ axis (This is the equivalent of sampling a random vertex along your instanced paths), so lets do that:



The points we just computed act as the first vertex of each edge, the other vertex can be computed mathematically or just by using the Project Point On Line node, the line starts are defined as the position of the next point and the ends are defined as the position of the next point plus some value in $x$ just to define the direction. To the location of the points of the next points, we simply shift the vector list backward, by doing so, each point is aligned with the one following it. Then we create the edges from the points and their projection. The node tree is as follows:



And the result is as expected (The horizontal lines are just for visual aid, not part of the implementation and they are the lines you originally had in your file):



You may notice that there is an extra edge going from the last vertex to the x axis, this is due to the fact that after shifting, the first vector became aligned with the last, and this is just a normal consequent of doing so. To fix that, just remove the last vector and last edge from the output using the slice  or remove list element node.